Final answer:
To find a vector parallel to v and a vector orthogonal to v such that their sum is equal to u, we can use the properties of parallel and orthogonal vectors.
Step-by-step explanation:
To find a vector parallel to v and a vector orthogonal to v such that their sum is equal to u, we can use the properties of parallel and orthogonal vectors.
Let's start by finding a vector parallel to v. Since parallel vectors have the same direction, we can simply multiply v by a scalar to get a vector parallel to it. Let's call this vector k*v, where k is a scalar. In this case, k*v = k<1,2,-1>.
Now let's find a vector orthogonal to v. We know that the dot product of orthogonal vectors is zero. So, to find a vector orthogonal to v, we can find a vector w such that the dot product of v and w is zero. Let's call this vector w. We can find w by solving the equation v · w = 0, where v = <1,2,-1>. If we replace w with , we get the equation 1*x + 2*y + (-1)*z = 0. By choosing appropriate values for x, y, and z, we can find a vector w that is orthogonal to v.
Now we can find the sum of k*v and w by adding their corresponding components. We have:
(k<1,2,-1>) + (w< x,y,z>) = <8,4,-12>
Solving this equation will give us the values of the scalar k and the components of vector w.